Related papers: The M\"obius Function of the Affine Linear Group $…

The M\"obius function of the subgroup lattice of a finite group has been introduced by Hall and applied to investigate several questions. In this paper, we consider the M\"obius function defined on an order ideal related to the lattice of…

Let $\mathbb{F}_q$ be the finite field with $q$ elements and consider the $n$-dimensional $\mathbb{F}_q$-vector space $V=\mathbb{F}_q^n\,$. In this paper we define a closure operator on the subgroup lattice of the group $G =…

We compute the M\"obius function for the subgroup lattice of the simple Suzuki group $Sz(q)$; this is used to enumerate normal subgroups of certain Hecke groups with quotients isomorphic to $Sz(q)$.

Let $n$ be a positive integer and let $\mathbb F_{q^n}$ be the finite field with $q^n$ elements, where $q$ is a power of a prime. This paper introduces a natural action of the Projective Semilinear Group $\text{P}\Gamma \text{L}(2,…

Let $B$ be a finite Boolean algebra. Let $\mathcal A$ be the partial order of all implication sublattices of $B$. We will compute the M\"obius function on $\mathcal A$ in two different ways.

The M\"{o}bius function of the subgroup lettice of a finite group $G$ has been introduced by Hall and applied to investigate several different questions. We propose the following generalization. Let $A$ be a subgroup of the automorphism…

Let $G$ be the simple group ${\rm PSL}(3,2^p)$, where $p$ is a prime number. For any subgroup $H$ of $G$, we compute the M\"obius function of $H$ in the subgroup lattice of $G$. To this aim, we describe the intersections of maximal…

We determine when an arithmetic subgroup of a reductive group defined over a global function field is of type FP_\infty by comparing its large-scale geometry to the large-scale geometry of lattices in real semisimple Lie groups.

Let $\text{AGL}(1,\Bbb F_q)$ be the affine linear group of dimension $1$ over a finite field $\Bbb F_q$. $\text{AGL}(1,\Bbb F_q)$ acts sharply 2-transitively on $\Bbb F_q$. Given $S<\text{AGL}(1,\Bbb F_q)$ and an integer $k$ with $1\le k\le…

In this paper the concept of $\mathbb{F}$-functorial of a finite group was introduced. These functorials have many properties of the Fitting subgroup of a soluble group and the generalized Fitting subgroup of a finite group. It was shown…

We show that a large class of divisible abelian $\ell$-groups (lattice ordered groups) of continuous functions is interpretable (in a certain sense) in the lattice of the zero sets of these functions. This has various applications to the…

Let $G$ be a finite group, $\mu$ be the M\"obius function on the subgroup lattice of $G$, and $\lambda$ be the M\"obius function on the poset of conjugacy classes of subgroups of $G$. It was proved by Pahlings that, whenever $G$ is…

Let $\text{U}(n,\mathbb{F}_{q^2})$ denote the subgroup of unitary matrices of the general linear group $\text{GL}(n,\mathbb{F}_{q^2})$ which fixes a Hermitian form and $M\geq 2$ an integer. This is a companion paper to the previous works…

The affine line and the punctured affine line over a finite field F are taken as benchmarks for the problem of describing geometric \'etale fundamental groups. To this end, using a reformulation of Tannaka duality we construct for a…

In this paper, we study the sum of additive characters over finite fields, with a focus on those of specified \(\mathbb{F}_q\)-Order. We establish a general formula for these character sums, providing an additive analogue to classical…

The principle result of this article is the determination of the possible finite subgroups of arithmetic lattices in U(2,1).

We prove that the M\"obius function is orthogonal to polynomials over $\mathbb{F}_q[x]$ (up to a characteristic condition). We use this orthogonality property to count prime solutions to affine-linear equations of bounded complexity in…

In this paper, we determine the genus of the subgroup lattice of several families of abelian groups. In doing so, we classify all finite abelian groups whose subgroup lattices can be embedded into the torus.

A unital $\ell$-group $(G,u)$ is an abelian group $G$ equipped with a translation-invariant lattice-order and a distinguished element $u$, called order-unit, whose positive integer multiples eventually dominate each element of $G$. We…

We give a realization of the quantum affine Lie superalgebras U_q(A(M,N))^(1) in terms of anyons defined on a one or two-dimensional lattice, the deformation parameter q being related to the statistical parameter $\nu$ of the anyons by q =…